Chapter 4: Clustering and classification
Henri V
2023-11-18
Chapter 4: Clustering & Classification
Data wrangling
Not included, as this time it is listed as the final assignment and deals with next week’s data. This has been completed; create_human.R” is available in the repo under Scripts.
Installation of the necessary packages
require(tidyverse);require(here);require(MASS);require(corrplot); require(GGally)Analysis
Data overview
- First, we load the Boston dataset and check its dimensions & structure.
data(Boston)
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14Overall the dimensions are 506 x 14.
All of the 14 variables appear numerical (integer or double).
Next, we look up what each of the 14 vars represents.
The dataset has an official documentation.
- Input command “?Boston” to view.
- In the documentation, variable definitions are provided.
The data contains information on housing in the Boston region, USA.
Included variables are:
- CRIM - per capita crime rate by town
- ZN - proportion of residential land zoned for lots over 25,000 sq.ft.
- INDUS - proportion of non-retail business acres per town.
- CHAS - Charles River dummy variable (1 if tract bounds river; 0 otherwise)
- NOX - nitric oxides concentration (parts per 10 million)
- RM - average number of rooms per dwelling
- AGE - proportion of owner-occupied units built prior to 1940
- DIS - weighted distances to five Boston employment centres
- RAD - index of accessibility to radial highways
- TAX - full-value property-tax rate per $10,000
- PTRATIO - pupil-teacher ratio by town
- B - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
- LSTAT - % lower status of the population
- MEDV - Median value of owner-occupied homes in $1000’s
Next, graphical and table format summaries are generated for the data
First, summary as a table
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00- The summary values for each var are wildly different. They are all on different scales, despite being all numeric. This is why standardization over vars will soon be done, to make them comparable.
ggpairs(Boston)
There’s quite a lot of options on what to look at here. I’m going to cherry pick some findings, instead of going through every variable.
Multiple variables have skewed distributions. For example:
- Age skews strongly towards high values -> mainly old buildings (built before -1940)
- Variable dis (distance to employment centers) has a skew towards low values (distance to employment centers commonly small)
Also, several variables have bimodality (= more than 1 peak in the distribution, meaning that 2 values are more common than others)
Scatterplot Indus x NOx appears to show 2 distinct groupings? For most data points, both NOX concentrations and the share of industrial acreage are low (these have a strong, statistically significant correlation coeff too!)
Crime rate appears to have a statistically significant correlation with almost all of these vars. Seems to correlate positively with the proportion of industrial acreage, NOX concentrations, large residential land areas…
The distribution of “indus” (business acreage) shows bimodality: we have two peaks, indicating that a couple of values are considerably more common than others. This variable also correlates w. high statistical significance with NOX emissions, which makes sense as the variable represents the prevalence of business acreage like industry.
Distribution of NOX is strongly skewed towards small values.
Age skews strongly towards high values. Overall, most construction in the regions of the data was done prior to 1940.
Again, we see bimodality in property tax rates. Low and high ends of the spectrum have clear peaks.
Dataset standardization
- As explained above, all these variables are numeric but have wildly different measurement scales. Hence, standardization.
- Let’s print summaries of both standardized and non-standardized data and compare
boston_scaled <- as.data.frame(scale(Boston))
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00- This procedure changed the scales on which all the different vars are measured. Previously, they were all different as they describe very different things. Now, we have “forced” all of them to a similar scale. See for example how the scale of “black” changes: max was almost 400, way more than for any other var. Due to standardization, it became 0.44. All the vars are now on the same scale.
Categorigal crime variable creation
- Division to bins according to quantiles, set it as a new variable to the old frame
bins <- quantile(boston_scaled$crim)
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
boston_scaled <- data.frame(boston_scaled, crime)
boston_scaled$crim<-NULL*Dividing the data to training set and testing set
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = nrow(boston_scaled) * 0.8)
# create train set and test set
train <- boston_scaled[ind,] #Randomly selects 80% of rows, index numbers
test <- boston_scaled[-ind,] #everything except the indices in the training setLDA
- Fitting LDA on the training data, then plotting it.
- Here’ we seek to find a linear combination of variables that best separates the data into groups.
- Looking at vector directions & magnitudes tells us about which
LD it has the most effect on
- Variables’ relationships with each other can also be seen. Parallel vectors show correlation, while perpendicular directions are the opposite case.
lda.fit <- lda(crime ~ . , data = train)
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale =2)
- RAD has the largest impact on LD1. For LD2 it lookse to be either
Nox or zn, albeit the magnitude appears similar…
- Nox and rad, as well as zn and rad, are almost perpendicular with each other (like a 90 degree angle) -> no correlation.
- Most of the other vars cluster very close together. None of them have vector magnitudes even close to those seen for zn, rad and nox.
Testing model prediction power
- First, we save the “real” class values of the test set, and remove it from the frame.
- We will use these to compare how well the same model we used on the training data classifies the actual test data-
correct_classes<-test$crime
test$crime<-NULL
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 17 9 4 0
## med_low 7 17 7 0
## med_high 0 6 15 2
## high 0 0 0 18- 30 correct “lows”, only 7 of which were correctly classified. 23% success rate…
- 25 med-lows, of which 12 are correctly classified. 48% success rate,
better but still not good enough.
- 26 med-highs, just 9 correctly placed. 34 %…
- 21 highs, of which 19 got correctly placed. A 90% success rate for this category -> high crime rates appear to be predicted very accurately, while in all other categories the performance is very lackluster.
K-Means
*Clearing .env, reload Boston raw data and scale it
data(Boston)
Boston_scaled<-as.data.frame(scale(Boston))*Calculating distances between data points
dist_eu <- dist(Boston_scaled)Run K-means clustering algorithm on the scaled data here, we run it on 6 centers as optimization will follow.
# k-means clustering
km <- kmeans(Boston_scaled, centers = "6")
pairs(Boston_scaled[1:6], col=km$cluster)
*Quite a confusing table…K of 6 is arguably too many.
*Optimizing K
- In the optimization, we look at the WCSS (Within group sum of squares)
- A high WCSS indicates that even though we have classified data into a same cluster group, there would still exist considerable variation within that group.
- We want to find a K value, for which the variation within each group is as small as possible (i.e. the clusters should be formed with the idea of grouping similar data points together, with as little “wild” variation between points in the same group as possible)
- Visually, we check the plotted curve, and observe at which point (K value) the most “benefit” has been gained from increasing K.
- The slope of WCSS is steep and rapidly decreases up to K-level of
circa 2. After that, we slope grows considerably milder -> even if we
increase K, the resulting “gains” (or here, losses, as we want small
WCSS values) grow less and less pronounced.
- So, after K= circa 2, not really worth it to add any more centers…
# Work with the exercise in this chunk, step-by-step. Fix the R code!
# MASS, ggplot2 and Boston dataset are available
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston_scaled, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
Re-running analysis with K set to 2
km <- kmeans(Boston_scaled, centers = 2)
pairs(Boston_scaled[1:5], col = km$cluster)
pairs(Boston_scaled[6:10], col = km$cluster)
- With K at 2, it’s a lot easier to see what’s happening.
- Let’s zoom into some of these:
- Multiple distinct groupings. For example:
- Black data points in the tax - age pairplot: cluster of properties high in both tax rate and age
pairs(Boston_scaled[c(1,10)], col = km$cluster)
- Crime rates and high property tax values (wealth indicator?) appear to cluster separately. Wealthier neighbourhoods, less crime?
pairs(Boston_scaled[c(3,5)], col = km$cluster)
- Industrial zoning and Nitrogen oxide emissions cluster as well. A distinct grouping of black data points show high emissions, when the rate of industrial zoning is also high. And when the zoning is not industrial, emissions are also low.
pairs(Boston_scaled[c(7,14)], col = km$cluster)
- Housing value and age appear to show a large cluster in black, with high age (old housing) and low value.
pairs(Boston_scaled[c(1,14)], col = km$cluster)
Bonus task
- Load original data, standardize, and run K means with 3 clusters
data("Boston")
Boston_scaled<-as.data.frame(scale(Boston))
km <- kmeans(Boston_scaled, centers = "5")- Then, fit LDA with clusters as the target (dependent)
lda.fit <- lda(km$cluster ~ . , data = Boston_scaled)
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
graphics::arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(km$cluster)
# plot the lda results (select both lines and execute them at the same time!)
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale =2.4)
#More zoom
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale =20)
- Now, location on the riverside looks to be the main determinant for group 1. For 3, it appears to be accessibility to radial highways. Remaining variables cluster together in a single mass, with no single ones standing out.
- In the more zoomed plot, LSTAT looks to be important for group 3, while zn is important for 2.
Super Bonus
- Install plotly
library(plotly)## Warning: package 'plotly' was built under R version 4.2.3##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout*Calculate matrix products
lda.fit <- lda(crime ~ . , data = train)
model_predictors <- dplyr::select(train, -crime)
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)3D Plot
Colored by the crime classes of the training data, we get the pink “High crime” data points grouped on their own (some overlap with med-high, though)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color= train$crime)Data wrangling for next week
- This has been completed; create_human.R” is available in the repo under Scripts and its output under Data.